{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:42:59Z","timestamp":1753893779287,"version":"3.41.2"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"2","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"<jats:p>We show that for every binary matroid $N$ there is a graph $H_*$ such that for the graphic matroid $M_G$ of a graph $G$, there is a matroid-homomorphism from $M_G$ to $N$ if and only if there is a graph-homomorphism from $G$ to $H_*$. With this we prove a complexity dichotomy for the problem $\\rm{Hom}_\\mathbb{M}(N)$ of deciding if a binary matroid $M$ admits a homomorphism to $N$. The problem is polynomial time solvable if $N$ has a loop or has no circuits of odd length, and is otherwise $\\rm{NP}$-complete. We also get dichotomies for the list, extension, and retraction versions of the problem.<\/jats:p>","DOI":"10.37236\/11119","type":"journal-article","created":{"date-parts":[[2023,5,19]],"date-time":"2023-05-19T13:45:16Z","timestamp":1684503916000},"source":"Crossref","is-referenced-by-count":0,"title":["The Complexity of the Matroid Homomorphism Problem"],"prefix":"10.37236","volume":"30","author":[{"given":"Cheolwon","family":"Heo","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Hyobin","family":"Kim","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Siggers","family":"Mark","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"23455","published-online":{"date-parts":[[2023,5,19]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v30i2p28\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v30i2p28\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,5,19]],"date-time":"2023-05-19T13:45:17Z","timestamp":1684503917000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v30i2p28"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,5,19]]},"references-count":0,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2023,4,7]]}},"URL":"https:\/\/doi.org\/10.37236\/11119","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2023,5,19]]},"article-number":"P2.28"}}